Multi-array laterolog tools and methods with differential voltage measurements

ABSTRACT

Multi-array laterolog tool systems and methods acquire a set of array measurements sufficient to provide laterolog tool measurements of differing array sizes. Such systems and method offer multiple depths of investigation while offering greater measurement stability in borehole environments having high resistivity contrasts. In at least some system embodiments, a wireline or LWD tool body has a center electrode positioned between multiple pairs of guard electrodes and a pair of return electrodes. The tool&#39;s electronics provide a current from the center electrode to the pair of return electrodes and currents from each pair of guard electrodes to the pair of return electrodes. Each of the currents may be distinguishable by frequency or distinguishable by some other means. This novel arrangement of currents provides a complete set of measurements that enables one tool to simultaneously emulate a whole range of laterolog tools.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of PCT/US2010/056645, titled“Multi-Array Laterolog Tools and Methods”, and filed Nov. 15, 2010, byInventors Michael S. Bittar, Shanjun Li, and Jing Li.

BACKGROUND

Modern oil field operators demand access to a great quantity ofinformation regarding the parameters and conditions encountereddownhole. Such information typically includes characteristics of theearth formations traversed by the borehole and data relating to the sizeand configuration of the borehole itself. The collection of informationrelating to conditions downhole, which commonly is referred to as“logging,” can be performed by several methods including wirelinelogging and “logging while drilling” (LWD).

In wireline logging, a sonde is lowered into the borehole after some orall of the well has been drilled. The sonde hangs at the end of a longwireline cable that provides mechanical support to the sonde and alsoprovides an electrical connection between the sonde and electricalequipment located at the surface of the well. In accordance withexisting logging techniques, various parameters of the earth'sformations are measured and correlated with the position of the sonde inthe borehole as the sonde is pulled uphole.

In LWD, the drilling assembly includes sensing instruments that measurevarious parameters as the formation is being penetrated, therebyenabling measurements of the formation while it is less affected byfluid invasion. While LWD measurements are desirable, drillingoperations create an environment that is generally hostile to electronicinstrumentation, telemetry, and sensor operations.

Among the available wireline and LWD tools are a variety of resistivitylogging tools including, in particular, “array laterolog” tools. Suchtools typically include a central electrode around a tool body, withguard electrodes symmetrically spaced above and below the centralelectrode. The tool drives auxiliary currents between the guardelectrodes and the center electrode to “focus” the current from thecenter electrode, i.e., to reduce dispersion of the current from thecenter electrode until after the current has penetrated some distanceinto the formation. Generally speaking, a greater depth of investigationcan be achieved using more widely-spaced guard electrodes, but thevertical resolution of the measurements may suffer. Accordingly,existing tools employ multiple sets of guard electrodes at differentspacings from the central electrode to enable multiple depths ofinvestigation without unduly sacrificing vertical resolution. Laterologtools with one, two, three, and four sets of guard electrodes have beencreated. Though measurements of the simpler tools are conceptuallysubsets of the measurements provided by the more complex tools, inpractice the presence of the extra guard electrodes affects themeasurements of the complex tools, thereby making it difficult tocompare measurements from different tools.

DESCRIPTION OF THE DRAWINGS

The various disclosed embodiments are better understood when thefollowing detailed description is considered in conjunction with theaccompanying drawings, in which:

FIG. 1 shows an illustrative environment for logging while drilling(“LWD”);

FIG. 2 shows an illustrative environment for wireline logging;

FIG. 3 shows an illustrative environment for tubing-conveyed logging;

FIG. 4 is a block diagram of an illustrative multi-array laterolog tool;

FIG. 5 shows an illustrative multi-array laterolog tool;

FIG. 6 illustrates a current flow pattern for a laterolog tool with sixsets of guard electrodes;

FIGS. 7A-7E illustrates the current flow patterns that can be derivedfrom measurements of the FIG. 6 current flow pattern;

FIG. 8 illustrates one method for deriving the flow pattern of FIG. 7A;

FIG. 9 illustrates a derivation of the flow pattern of FIG. 7B; and

FIG. 10 is a flow diagram of a multi-array laterolog logging method.

While the invention is susceptible to various modifications andalternative forms, specific embodiments are shown by way of example inthe drawings and will herein be described in detail. It should beunderstood, however, that the drawings and detailed description are notintended to limit the disclosure, but on the contrary, the intention isto cover all modifications, equivalents and alternatives falling withinthe scope of the appended claims.

DETAILED DESCRIPTION

Accordingly, there are disclosed herein improved multi-array laterologtool systems and methods that acquire a set of array measurementssufficient to provide laterolog tool measurements of differing arraysizes. Such systems and method offer multiple depths of investigationwhile offering greater measurement stability in borehole environmentshaving high resistivity contrasts. In at least some system embodiments,a wireline or LWD tool body has a center electrode positioned betweenmultiple pairs of guard electrodes and a pair of return electrodes. Amonitor electrode dyad is positioned between the center electrode andinnermost guard electrodes, and similar monitor electrode dyads arepositioned between adjacent guard electrodes. The tool's electronicsprovide a current from the center electrode to the pair of returnelectrodes and currents from each pair of guard electrodes to the pairof return electrodes. The electronics further operate to acquiredifferential voltage measurements between at least one of the monitorelectrode dyads.

Each of the currents may be distinguishable by frequency ordistinguishable by some other means. This novel arrangement of currentsprovides a complete set of measurements that enables one tool tosimultaneously emulate a whole range of laterolog tools. Thecontemplated numbers of guard electrode pairs ranges from three to five,though of course more can be employed if space permits.

The disclosed systems and methods are best understood in the context ofthe larger environments in which they operate. Suitable environments areillustrated in FIGS. 1-3.

FIG. 1 shows an illustrative logging while drilling (LWD) environment. Adrilling platform 2 is equipped with a derrick 4 that supports a hoist 6for raising and lowering a drill string 8. The hoist 6 suspends a topdrive 10 suitable for rotating the drill string 8 and lowering the drillstring through the well head 12. Connected to the lower end of the drillstring 8 is a drill bit 14. As bit 14 rotates, it creates a borehole 16that passes through various formations 18. A pump 20 circulates drillingfluid through a supply pipe 22 to top drive 10, down through theinterior of drill string 8, through orifices in drill bit 14, back tothe surface via the annulus around drill string 8, and into a retentionpit 24. The drilling fluid transports cuttings from the borehole intothe pit 24 and aids in maintaining the integrity of the borehole 16.Various materials can be used for drilling fluid, including a salt-waterbased conductive mud.

A LWD tool suite 26 is integrated into the bottom-hole assembly near thebit 14. As the bit extends the borehole through the formations, logging,tool 26 collects measurements relating to various formation propertiesas well as the tool orientation and various other drilling conditions.The LWD tools 26 may take the form of a drill collar, i.e., athick-walled tubular that provides weight and rigidity to aid thedrilling process. (For the present discussion, the set of logging toolsis expected to include a multi-array laterolog resistivity tool tomeasure formation resistivity.) A telemetry sub 28 may be included totransfer images and measurement data to a surface receiver 30 and toreceive commands from the surface. In some embodiments, the telemetrysub 28 does not communicate with the surface, but rather stores loggingdata for later retrieval at the surface when the logging assembly isrecovered.

At various times during the drilling process, the drill string 8 may beremoved from the borehole as shown in FIG. 2. Once the drill string hasbeen removed, logging operations can be conducted using a wirelinelogging sonde 34, i.e., a probe suspended by a cable 42 havingconductors for transporting power to the sonde and telemetry from thesonde to the surface. A wireline logging sonde 34 may have pads and/orcentralizing springs to maintain the tool near the axis of the boreholeas the tool is pulled uphole. Logging sonde 34 can include a variety ofsensors including a multi-array laterolog tool for measuring formationresistivity. A logging facility 44 collects measurements from thelogging sonde 34, and includes a computer system 45 for processing andstoring the measurements gathered by the sensors.

An alternative logging technique is tubing-conveyed logging. FIG. 3shows an illustrative coil tubing logging system in which coil tubing 54is pulled from a spool 52 by a tubing injector 56 and injected into awell through a packer 58 and a blowout preventer 60 into the well 62. Inthe well, a supervisory sub 64 and one or more logging tools 65 arecoupled to the coil tubing 54 and configured to communicate to a surfacecomputer system 66 via information conduits or other telemetry channels.An whole interface 67 may be provided to exchange communications withthe supervisory sub and receive data to be conveyed to the surfacecomputer system 66.

Surface computer system 66 is configured to communicate with supervisorysub 64 to set logging parameters and collect logging information fromthe one or more logging tools 65 such as a multi-array laterolog tool.Surface computer system 66 is configured by software (shown in FIG. 3 inthe form of removable storage media 72) to monitor and control downholeinstruments 64, 65. System 66 includes a display device 68 and auser-input device 70 to enable a human operator to interact with thesystem control software 72.

In each of the foregoing logging environments, the logging toolassemblies may include a navigational sensor package having directionalsensors for determining the inclination angle, the horizontal angle, andthe rotational angle (a.k.a. “tool face angle”) of the bottomholeassembly (BHA). As is commonly defined in the art, the inclination angleis the deviation from vertically downward, the horizontal angle is theangle in a horizontal plane from true North, and the tool face angle isthe orientation (rotational about the tool axis) angle from the highside of the wellbore. In accordance with known techniques, directionalmeasurements can be made as follows: a three axis accelerometer measuresthe earth's gravitational field vector relative to the tool axis and apoint on the circumference of the tool called the “tool face scribeline”. (The tool face scribe line is typically drawn on the tool surfaceas a line parallel to the tool axis.) From this measurement, theinclination and tool face angle of the BHA can be determined.Additionally, a three axis magnetometer measures the earth's magneticfield vector in a similar manner. From the combined magnetometer andaccelerometer data, the horizontal angle of the BRA may be determined.

A discussion of the multi-array laterolog tool electronics is in orderbefore describing the physical construction of the tool. FIG. 4 shows afunctional block diagram of the tool electronics. The control module 410governs the operation of the tool in accordance with software and/orfirmware 412 stored in internal memory. The control module 410 couplesto telemetry module 420 to receive commands and to provide measurementdata. Control module 410 further connects to digital-to-analog converter430 to drive current electrodes 432, and connects to analog-to-digitalconverter 440 to make voltage measurements via monitor electrodes 442.Control module 410 can be, for example, a general purpose processor, adigital signal processor, a programmable gate array, or an applicationspecific integrated circuit. Telemetry module 420 receives and storesmeasurement data in a nonvolatile memory 422, and further operates as acommunications interface between the control module 410 and thetelemetry communications mechanism.

FIG. 4 shows 2N+1 current electrodes (electrodes A₀, A₁, A₂, . . .A_(N), A₁′, A₂′, . . . A_(N)′) being independently driven viadigital-to-analog converter 430. In some tool embodiments, the guardelectrodes are electrically connected in pairs, i.e., electrode A₁ isconnected to electrode A₁′ by an electrical conductor, electrode A₂ isconnected to electrode A₂′, etc. Moreover, the return electrodes areelectrically connected (i.e., electrode A_(N) is conductively coupled toelectrode A_(N)′). In such alternative embodiments, thedigital-to-analog converter can be simplified to drive only oneelectrode in each pair. Similarly, the monitor electrodes 442 can beelectrically connected in pairs, i.e., with electrode M₁ connected toM₁′, electrode M₂ connected to M₂′, etc. It is also contemplated thateach electrode can be individually driven/sensed and that the controlmodule can collect the pair-wise measurements by appropriately combiningthe individual electrode currents and voltages.

A series of differential amplifiers 441 provides the analog to digitalconverter 440 of FIG. 4 with differential voltage measurements betweenmonitor electrode dyads, e.g., between monitor electrodes M₁ and M₂,between M₃ and M₄, . . . , and between M_(2N-3) and M_(2N-2). Where themonitor electrodes are not connected in pairs, the analog to digitalconverter 440 further measures the voltages between monitor electrodesM₁′ and M₂′, between M₃′ and M₄′, . . . , and between M_(2N-3)′ andM_(2N-2)′. For completeness, the analog, to digital converter 440 mayalso measure the voltages of the odd-numbered monitor electrodes (M₁,M₃, . . . , M_(2N-3), M₁′, M₃′, . . . , M_(2N-3)′). Othernon-differential measurements could alternatively be used, such as thevoltages of the even-numbered monitor electrodes, or the average voltagefor each monitor electrode dyad. Given both the differential andnon-differential measurements, the tool can determine the voltage foreach monitor electrode.

The acquisition of differential measurements is desirable because suchmeasurements are, in many cases, very small relative to thenon-differential voltages. Moreover, the derived resistivity can be verysensitive to error in the differential values, so it is desirable toacquire these measurements with a dedicated, high accuracyanalog-to-digital converter rather than digitizing the monitor electrodevoltages separately before determining the differences.

FIG. 5 shows an illustrative multi-array laterolog tool 502 having (forthe sake of illustration) equally-spaced current electrodes and returnelectrodes (electrodes A₀, A₁, A₂, . . . A₆, A₁′, A₂′, . . . A₆′), withinterspersed monitor electrodes M₁-M₁₀ and M₁′-M₁₀′ on a wireline toolbody. (It is desirable to separate the monitor electrodes from thecurrent electrodes because the current electrodes often develop animpedance layer that distorts voltage measurements when current isflowing.) This disclosure often groups the monitor electrodes into pairsand dyads. The term “pair” will be consistently used to refer to monitorelectrodes symmetrically located relative to the center electrode, e.g.M₁ and M₁′, or M₆ and M₆′. The term “dyad” will be consistently used torefer to the two monitor electrodes between adjacent current electrodes(e.g., M₁ and M₂, or M₅′ and M₆′) or to the corresponding two monitorelectrode pairs (e.g., the pair M₁ and M₁′, taken with the pair M₂ andM₂′, form a dyad of monitor electrode pairs).

Though the figure shows equally-spaced, uniformly-sized currentelectrodes, the electrodes are typically not equally sized and spaced.Better performance is achieved by having the more distant electrodesincrease in size. Thus, in one contemplated embodiment the centerelectrode A₀ has an axial length of 6 inches. The lengths of electrodesA_(i) and A_(i)′ for i ranging from 1 to 6 is (in inches) 6, 8, 10, 14,20, and 75. The spacing between the current electrodes also increases,beginning at 6 inches between electrodes A₀ and A₁, 6 inches betweenelectrodes A₁ and A₂, 10 inches between electrodes A₂ and A₃, 14 inchesbetween A₃ and A₄, 19 inches between A₄ and A₅, and 34 inches between A₅and A₆. (These spacings are measured between the nearest edges and notcenter to center, and they are symmetric with respect to the centerelectrode.) In this contemplated embodiment, each of the monitorelectrodes has an axial length of 1 inch. With one exception, themonitor electrodes are spaced 1 inch away from the nearest currentelectrode. (Electrodes M₂ and M₂′ may be spaced 2 inches from currentelectrodes A₁ and A₁′, respectively.)

The tool electronics employ the current electrodes to provide thecurrents I₀-I₅ and I′₀-I₅′ as illustrated in FIG. 6. Currents I₀-I₅ aresourced from electrodes A₀-A₅ respectively, with electrode A₆ serving asa common return electrode for each of these currents. Similarly,currents I₀′-I₅′ are sourced from electrodes A₀ and A₁′-A₅′respectively, with electrode A₆′ serving as a common return electrodefor these currents. If the current and monitor electrodes are pair-wiseconnected as discussed before, the tool cannot distinguish currentsI₀-I₅ from I₀′-I₅′, but instead operates on the combined currents(I₀+I₀′, I₁+I₁′, I₂+I₂′, . . . ). Otherwise, the tool can analyzecurrents I₀-I₅ separately from I₀′-I₅′, or in the alternative, combinethe currents and voltages digitally before analyzing. Due to thesymmetry of the array, only the top half is illustrated in the ensuingfigures. Though not shown, the bottom half is presumed to be present.

To enable the monitor electrodes to distinguish the effects of thevarious currents, the currents are given distinguishable features. Inthe contemplated tool embodiment, the electrodes are pair-wise connectedand currents I₀-I₅ have distinguishable signal frequencies f₁-f₆. Thecontemplated set of frequencies includes 80 Hz, 112 Hz, 144 Hz, 176 Hz,208 Hz, and 272 Hz. (It is expected that the borehole fluid will befairly conductive, thereby allowing low frequency currents to pass intoand through the formation.) This frequency set offers sufficientfrequency spacing to enable fast logging, while not spreading thefrequencies so far apart as to incur excessive frequency dependence inthe resistivity measurements. Moreover this frequency set avoids the useof harmonic frequencies which could be unduly sensitive to nonlineareffects in the system. Nevertheless, other sets of frequencies wouldalso be suitable for distinguishing the currents. Alternatively, thecurrents could be distinguished through the use of time divisionmultiplexing, code division multiplexing, or other methods that enablethe currents to be independently monitored.

While each of the currents is provided with a characteristic that makesits effects distinguishable from those of the other currents, in atleast some tool embodiments some of the currents are given commonfeatures. For example, some tool embodiments provide current I₀ withfrequencies f₁ and f₂. The sharing of frequency f₂ by both current I₀and I₁ enables straightforward hardware focusing as described in greaterdetail below.

As the tool drives the current electrodes, the currents pass through theborehole fluid and the formation to reach the return electrodes,creating a field potential indicative of the resistivity of thematerials along the various current flow paths. The control modulerecords a voltage signal from each monitor electrode to measure thefield potential at the monitor electrode locations. A frequency analysisof the voltage signals (e.g., by Fourier transform, filtering, orleast-squares curve fitting) separates out those voltage signalcomponents attributable to each of the currents.

With the measurements for the current flow pattern of FIG. 6, it becomespossible to derive the measurements associated with each of the currentflow patterns provided in FIGS. 7A-7E, FIG. 7E represents the full arraymeasurement (which corresponds to the actual current flow pattern ofFIG. 6), while FIGS. 7A-7D represent truncated array measurements ofvarious degrees. In FIG. 7A (sometimes referred to below as Mode 1),current electrode A₂ is the shared return electrode, whereas in FIG. 7B(Mode 2), current electrode A₃ is the shared return electrode, and soon. By determining measurements for each of the array sizes, the toolcan provide resistivity measurements not only as a function of toolposition, but also as a function of radial distance from the borehole.

FIG. 8 demonstrates one approach for deriving the tool measurements ofFIG. 7A (Mode 1). From the measurements for the complete set of currents802, the measurements corresponding to currents I₀, I₁, and I₂ areextracted label 804), e.g., by identifying those components of themonitor electrode voltage signals having the corresponding frequencycomponents. The difference between the measurements for the desiredtruncated current flow pattern 808 and the extracted measurements 804 isthat set of voltage measurements that would be obtained in response tothe current flow pattern 806, which can be readily derived from themeasurements corresponding to current I₂. Representing the extractedvoltages for monitor electrode M_(i) and voltages difference for monitorelectrode dyad M_(i) and M_(k) in vector form:v _(i) =[v _(i,1) v _(i,2) v _(i,3)]  (1)Δv _(i,k) =[Δv _(i,k,1) Δv _(i,k,2) Δv _(i,k,3)]  (2)where the different vector components correspond to differentfrequencies f₁, f₂, f₃. (Throughout the following description, v_(i,j)represents the jth frequency component of the voltage signal received bythe ith monitor electrode, and Δv_(i,k,j) represents the jth frequencycomponent of the voltage difference signal received between ith and kthmonitor electrodes.) If the currents I₀, I₁, and I₂ differ only infrequency and not magnitude, then the truncated flow patternmeasurements 808 are:v _(i) ′=[v _(i,1) −v _(i,3) v _(i,2) −v _(i,3) v _(i,3) −v _(i,3)]  (3)The last vector component is of course zero, as I₂ is not part of thetruncated flow pattern. (Where the current magnitudes are not equal themeasurements should be scaled accordingly before applying thecorrection.)

Those familiar with laterolog tools recognize that the analysis is notyet complete, as the tool has not yet provided for focusing of thecurrent. As with existing laterolog tools, focusing is accomplished bybalancing the current from the center electrode with currents from theguard electrodes. In the current flow pattern of FIG. 7A, the properbalance had been achieved when monitor electrodes M₁ and M₂ have signalswith equal contributions from each current, i.e., when:v′ _(1,1) −v′ _(2,1) =−c(v′ _(1,2) −v′ _(2,2))  (4)where c is an internal scale factor that causes I₁ to balance I₀. (Thisuse of internal scale factors is herein teemed “software focusing”.) Theapparent resistivity is then

$\begin{matrix}{R_{{Model}\; 1} = {k_{1}\frac{V_{M\; 1}}{I_{0}}}} & (5)\end{matrix}$where k₁ is a tool constant, I₀ is the current from the centerelectrode, andV _(M1) =v′ _(1,1) +cv′ _(1,2).  (6)

The foregoing approach can be condensed into the following equations

$\begin{matrix}{{{\begin{bmatrix}{\Delta\; v_{1,2,2}} & \Delta_{1,2,3} \\I_{1,2} & I_{2,3}\end{bmatrix}\begin{bmatrix}c_{1,1} \\c_{1,2}\end{bmatrix}} = {- \begin{bmatrix}{\Delta\; v_{1,2,1}} \\I_{0,1}\end{bmatrix}}},} & (7)\end{matrix}$which is solved to find c_(1,1) and c_(1,2), andV _(M1) =v _(1,1) +c _(1,1) v _(1,2) +c _(1,2) v _(1,3),  (8)in combination with equation (5) above. The first and second subscriptsin I_(k,j) are the source electrode (A₀, A₁, . . . ) and the frequency(f₁, f₂, . . . ). The second subscript is added for generality. Certaintool embodiments employ currents with multiple frequency components toenable adaptive hardware balancing of the currents. For example, thecurrent from electrode A₀ can include two signal frequencies f₁ and f₂,where f₂ is also the signal frequency of the current from electrode A₁.The control module 410 (FIG. 4) can then dynamically adjust the relativemagnitude of currents I0 and I1 to ensure that v_(1,2)=v_(2,2), atechnique which is herein termed “hardware focusing”. Hardware focusingis expected to improve measurement stability. Nevertheless, to obtainthe measurements of a truncated array laterolog, software focusing isstill used. In this example the equations become:

$\begin{matrix}{{{\begin{bmatrix}{\Delta\; v_{1,2,1}} & {\Delta\; v_{1,2,3}} \\I_{0,1} & I_{2,3}\end{bmatrix}\begin{bmatrix}c_{1,1} \\c_{1,2}\end{bmatrix}} = {- \begin{bmatrix}{\Delta\; v_{1,2,2}} \\{I_{0,2} + I_{1,2}}\end{bmatrix}}},} & (9)\end{matrix}$withV _(M1) =c _(1,1) v _(1,1) +v _(1,2) +c _(1,3) v _(1,3).  (10)andI ₀ =c _(1,1) I _(0,1) +I _(0,2)  (11)in combination with equation (5) above.

Thus suitable equations for the software focusing approach and thecombined hardware/software focusing approach for Mode 1 have beendisclosed. FIG. 9 demonstrates a similar approach for deriving the toolmeasurements of FIG. 7B (Mode 2). From the measurements for the completeset of currents 802, the measurements corresponding to currents I₀, I₁,I₂, I₃ are extracted (label 904), e.g., by identifying those componentsof the monitor electrode voltage signals having the correspondingfrequency components and by identifying those components of the monitorelectrode. With the correction representing the current flow pattern906, the truncated current flow measurements 908 can be found. Theequations for the software-focused approach are:

$\begin{matrix}{R_{{Model}\; 2} = {k_{2}\;\frac{V_{M\; 1}}{I_{0}}}} & (12)\end{matrix}$where k₂ is the tool constant for Mode 2 andV _(M1) =v _(1,1) +c _(2,1) v _(1,2) +c _(2,2) v _(1,3) +c _(2,3) v_(1,4),  (13)with the coefficients being the solution to the simultaneous set ofequations

$\begin{matrix}{{\begin{bmatrix}{\Delta\; v_{1,2,2}} & {\Delta\; v_{1,2,3}} & {\Delta\; v_{1,2,4}} \\\begin{matrix}{{\Delta\; v_{3,4,2}} -} \\{\left( {\lambda_{2,2} - 1} \right)v_{4,2}}\end{matrix} & \begin{matrix}{{\Delta\; v_{3,4,3}} -} \\{\left( {\lambda_{2,2} - 1} \right)v_{4,3}}\end{matrix} & \begin{matrix}{{\Delta\; v_{3,4,4}} -} \\{\left( {\lambda_{2,2} - 1} \right)v_{4,4}}\end{matrix} \\I_{1,2} & I_{2,3} & I_{3,4}\end{bmatrix}\begin{bmatrix}c_{2,1} \\c_{2,2} \\c_{2,3}\end{bmatrix}} = {- {\begin{bmatrix}{\Delta\; v_{1,2,1}} \\\begin{matrix}{{\Delta\; v_{3,4,1}} -} \\{\left( {\lambda_{2,2} - 1} \right)v_{4,1}}\end{matrix} \\I_{0,1}\end{bmatrix}.}}} & (14)\end{matrix}$This last set of equations introduces the use of an enhanced focusingfactor λ_(m,n), which is a desired ratio between the voltages onselected monitor electrodes. The first subscript m is the mode number (2indicates Mode 2) and the second subscript n is the position of themonitor electrode dyad (n=2 indicates the monitor electrode dyad betweencurrent electrodes A₁ and A₂, whereas n=3 indicates the monitorelectrode dyad between A₂ and A₃, n=4 indicates the monitor electrodedyad between A₃ and A₄, and n=5 indicates the monitor electrode dyadbetween A₄ and A₅). The focusing factor is a selected ratio between theinner monitor electrode and the outer monitor electrode (e.g., λ_(2,2)is the selected ratio of M₃ to M₄). By default, the selected value of λis 1.0 (meaning enforced equality), but it has been found that often adeeper depth of investigation can be achieved by decreasing λ by around10 to 20 percent. Note that to make the ensuing equations more compact,we abbreviate the quantity (λ_(m,n)−1) as λ_(m,n) ^(c).

For the combined hardware/software focusing approach (where the toolsources I₀ with two frequency components as described previously),equation (11) is used in combination with the following equations:V _(M1) =c _(2,1) v _(1,1) +V _(1,2) +c _(2,2) v _(1,3) +c _(2,3) v_(1,4),  (14)I ₀ =c _(2,1) I _(0,1) +I _(0,2)  (15)with the coefficients being, the solution to the simultaneous set ofequations

$\begin{matrix}{{\begin{bmatrix}{\Delta\; v_{1,2,1}} & {\Delta\; v_{1,2,3}} & {\Delta\; v_{1,2,4}} \\\begin{matrix}{{\Delta\; v_{3,4,1}} -} \\{\lambda_{2,2}^{c}v_{4,1}}\end{matrix} & \begin{matrix}{{\Delta\; v_{3,4,3}} -} \\{\lambda_{2,2}^{c}v_{4,3}}\end{matrix} & \begin{matrix}{{\Delta\; v_{3,4,4}} -} \\{\lambda_{2,2}^{c}v_{4,4}}\end{matrix} \\I_{0,1} & I_{2,3} & I_{3,4}\end{bmatrix}\begin{bmatrix}c_{2,1} \\{\; c_{2,2}} \\c_{2,3}\end{bmatrix}} = {- {\begin{bmatrix}{\Delta\; v_{1,2,2}} \\{{\Delta\; v_{3,4,2}} - {\lambda_{2,2}^{c}v_{4,2}}} \\{I_{0,2} + I_{1,2}}\end{bmatrix}.}}} & (16)\end{matrix}$

For Mode 3 (FIG. 7C) the equations for the software focused approachare:

$\begin{matrix}{R_{{Model}\; 3} = {k_{3}\frac{V_{M\; 1}}{I_{0}}}} & (17)\end{matrix}$where k3 is the tool constant for Mode 3 andV _(M1) =v _(1,1) +c _(3,1) v _(1,2) +c _(3,2) v _(1,3) +c _(3,3) v_(1,4) +c _(3,4) v _(1,5),  (18)with the coefficients being the solution to the simultaneous set ofequations

$\begin{matrix}{{\begin{bmatrix}{\Delta\; v_{1,2,2}} & {\Delta\; v_{1,2,3}} & {\Delta\; v_{1,2,4}} & {\Delta\; v_{1,2,5}} \\\begin{matrix}{{\Delta\; v_{3,4,2}} -} \\{\lambda_{3,2}^{c}v_{4,2}}\end{matrix} & \begin{matrix}{{\Delta\; v_{3,4,3}} -} \\{\lambda_{3,2}^{c}v_{4,3}}\end{matrix} & \begin{matrix}{{\Delta\; v_{3,4,4}} -} \\{\lambda_{3,2}^{c}v_{4,4}}\end{matrix} & \begin{matrix}{{\Delta\; v_{3,4,5}} -} \\{\lambda_{3,2}^{c}v_{4,5}}\end{matrix} \\\begin{matrix}{{\Delta\; v_{5,6,2}} -} \\{\lambda_{3,3}^{c}v_{6,2}}\end{matrix} & \begin{matrix}{{\Delta\; v_{5,6,3}} -} \\{\lambda_{3,3}^{c}v_{6,3}}\end{matrix} & \begin{matrix}{{\Delta\; v_{5,6,4}} -} \\{\lambda_{3,3}^{c}v_{6,4}}\end{matrix} & \begin{matrix}{{\Delta\; v_{5,6,5}} -} \\{\lambda_{3,3}^{c}v_{6,5}}\end{matrix} \\I_{0,1} & I_{2,3} & I_{3,4} & I_{4,5}\end{bmatrix}\begin{bmatrix}c_{3,1} \\c_{3,2} \\c_{3,3} \\c_{3,4}\end{bmatrix}} = {- \begin{bmatrix}{\Delta\; v_{1,2,1}} \\\begin{matrix}{{\Delta\; v_{3,4,1}} -} \\{\lambda_{3,2}^{c}v_{4,1}}\end{matrix} \\\begin{matrix}{{\Delta\; v_{5,6,1}} -} \\{\lambda_{3,3,}^{c}v_{6,1}}\end{matrix} \\I_{0,1}\end{bmatrix}}} & (19)\end{matrix}$

For the combined hardware/software focusing approach (where the toolsources I₀ with two frequency components as described previously),equation (17) is used in combination with the following equations:V _(M1) =c _(3,1) v _(1,1) +V _(1,2) +c _(3,2) v _(1,3) +c _(3,3) v_(1,4) +c _(3,4) v _(1,5),  (20)I ₀ =c _(3,1) I _(0,1) +I _(0,2)  (21)with the coefficients being the solution to the simultaneous set ofequations

$\begin{matrix}{{\begin{bmatrix}{\Delta\; v_{1,2,1}} & {\Delta\; v_{1,2,3}} & {\Delta\; v_{1,2,4}} & {\Delta\; v_{1,2,5}} \\\begin{matrix}{{\Delta\; v_{3,4,1}} -} \\{\lambda_{3,2}^{c}v_{4,1}}\end{matrix} & \begin{matrix}{{\Delta\; v_{3,4,3}} -} \\{\lambda_{3,2}^{c}v_{4,3}}\end{matrix} & \begin{matrix}{{\Delta\; v_{3,4,4}} -} \\{\lambda_{3,2}^{c}v_{4,4}}\end{matrix} & \begin{matrix}{{\Delta\; v_{3,4,5}} -} \\{\lambda_{3,2}^{c}v_{4,5}}\end{matrix} \\\begin{matrix}{{\Delta\; v_{5,6,1}} -} \\{\lambda_{3,3}^{c}v_{6,1}}\end{matrix} & \begin{matrix}{{\Delta\; v_{5,6,3}} -} \\{\lambda_{3,3}^{c}v_{6,3}}\end{matrix} & \begin{matrix}{{\Delta\; v_{5,6,4_{4}}} -} \\{\lambda_{3,3}^{c}v_{6,4}}\end{matrix} & \begin{matrix}{{\Delta\; v_{5,6,5}} -} \\{\lambda_{3,3}^{c}v_{6,5}}\end{matrix} \\I_{0,1} & I_{2,3} & I_{3,4} & I_{4,5}\end{bmatrix}\begin{bmatrix}c_{3,1} \\c_{3,2} \\c_{3,3} \\c_{3,4}\end{bmatrix}} = {- \begin{bmatrix}{\Delta\; v_{1,2,2}} \\\begin{matrix}{{\Delta\; v_{3,4,2}} -} \\{\lambda_{3,2}^{c}v_{4,2}}\end{matrix} \\\begin{matrix}{{\Delta\; v_{5,6,2}} -} \\{\lambda_{3,3}^{c}v_{6,2}}\end{matrix} \\{{I_{0,2} + I_{1,2}}\;}\end{bmatrix}}} & (22)\end{matrix}$

The equations for Modes 4 and 5 can be readily written by observing thepatterns made apparent in the equations for Modes 1-3. Each column inthe simultaneous-equation matrices corresponds to a signal frequency.The last row corresponds to the current components, while each of theremaining rows is the desired balance for the two monitor electrodesbetween a given set of current electrodes. Where the focusing factor is1.0 (i.e., λ^(c)=0), the entries for all but the last row are providedby a frequency analysis of the differential voltage measurementsacquired by the A/D converter. Where the focusing factor is slightlyless than one (e.g., λ^(c)=0.1), the non-differential measurementscontribute to these rows. Even where the focusing factor is reduced to0.8 (i.e., λ^(c)=0.2), the error contribution of the non-differentialmeasurement is substantially reduced relative to a calculation thatrelies solely on non-differential measurements.

FIG. 10 provides an overview of a multi-array laterolog resistivitylogging method. Beginning in block 1402, the tool is conveyed through aborehole containing a conductive fluid. The tool can be drawn throughthe borehole by a wireline cable, or conveyed on a tubing string, orincorporated into the bottom hole assembly of a drill string. In block1404 the tool electronics energize the current electrodes to generatecurrents in the manner outlined previously. In block 1406, the tool orsome other component of the system tracks the motion and/or position ofthe tool as the tool electronics sample the differential andnon-differential voltage signals from the monitor electrodes. In block1408, the tool electronics record the voltage signals into aninformation storage medium and/or communicate the voltage signal data toa processing facility on the surface. In block 1410, the voltage signalsare processed (downhole or at the surface) in accordance with one of themethods discussed above to determine the monitor electrode measurementsand/or the generated currents (e.g., V_(M1) and/or I₀) expected for eachof the tool modes 1-5 (see FIGS. 7A-7E). In block 1412, the voltage andcurrent values for the various modes are used to determine formationresistivity measurements at different depths of investigation (i.e.,different effective radial measurement depths), enabling the loggingsystem to determine a formation resistivity log that depends both onposition along the borehole axis and on radial distance from theborehole axis. Some or all of the formation resistivity log data isdisplayed to a user in block 1414.

Numerous variations and modifications will become apparent to thoseskilled in the art once the above disclosure is fully appreciated. It isintended that the claims be interpreted to embrace ail such variationsand modifications.

What is claimed is:
 1. A resistivity logging system that comprises: atool body having: a center electrode positioned between multiple pairsof guard electrodes, wherein said multiple pairs of guard electrodes arepositioned between a pair of return electrodes on said tool body, eachsaid pair being a set of two electrodes symmetrically located relativeto the center electrode; and between each adjacent pair of guardelectrodes, a dyad of monitor electrode pairs, each dyad being twomonitor electrode pairs that form a set of four monitor electrodessymmetrically located relative to the center electrode, the set of fourmonitor electrodes including, on each side of the center electrode, twomonitor electrodes between adjacent guard electrodes in the adjacentpair of guard electrodes; and electronics that provide a primary currentfrom the center electrode to the pair of return electrodes and provide arespective guard current from each pair in said multiple pairs of guardelectrodes to the pair of return electrodes, the electronics furtheroperating to acquire a differential voltage measurement from at leastone dyad of monitor electrode pairs.
 2. The system of claim 1, whereinsaid multiple pairs of guard electrodes includes at least three pairs,and wherein the electronics further operate to acquire a differentialvoltage measurement from each dyad of monitor electrode pairs.
 3. Thesystem of claim 2, wherein the tool body further includes a dyad ofmonitor electrode pairs between the center electrode and an innermostpair of guard electrodes.
 4. The system of claim 1, wherein theelectronics drive all of said primary and guard currents concurrently.5. The system of claim 4, wherein the primary current and each of therespective guard currents has a spectral signature that distinguishes itfrom other currents.
 6. The system of claim 4, wherein each of the guardcurrents has a unique signal frequency.
 7. The system of claim 6,wherein the primary current has at least one signal frequency differentthan each of the guard current signal frequencies.
 8. The system ofclaim 7, wherein the primary current further includes a signal frequencythat matches a signal frequency of an innermost pair of guardelectrodes.
 9. The system of claim 1, wherein the guard electrodes ineach pair are shorted together, and the monitor electrodes in each pairare shorted together.
 10. The system of claim 1, further comprising: aprocessor that receives that receives measurements indicative ofelectrical resistances associated with the primary current and the guardcurrents, wherein the processor processes the measurements to determinemeasurements corresponding to a tool having a reduced number of guardelectrodes.
 11. The system of claim 1, wherein the tool body is adrilling collar.
 12. The system of claim 1, wherein the tool body is awireline sonde.
 13. A resistivity logging method that comprises:receiving from a logging tool measurements indicative of a differentialvoltage between each dyad of monitor electrode pairs positioned betweenadjacent guard electrode pairs, each pair being a set of two electrodeslocated symmetrically relative to a center electrode and each dyad beingtwo monitor electrode pairs that form a set of four monitor electrodessymmetrically located relative to the center electrode, the set of fourmonitor electrodes including, on each side of the center electrode, twomonitor electrodes between adjacent guard electrodes in each adjacentpair of guard electrodes; processing said measurements to determine alocalized formation resistivity at a position associated with a locationof the center electrode during collection of said measurements; anddisplaying said localized formation resistivity as a function ofposition.
 14. The method of claim 13, wherein said processing comprises:deriving measurements indicative of electrical resistances between thecenter electrode and each pair of guard electrodes; and combining saidderived measurements in a manner that enforces an assumption ofcommensurate voltages on the center electrode and at least one pair ofguard electrodes.
 15. The method of claim 13, further comprising drivingsaid guard electrode pairs and center electrode with respective currentsources.
 16. The method of claim 15, wherein the respective currentsources provide currents with different signal frequencies.
 17. Aresistivity logging tool that comprises: a tool body having a centerelectrode positioned between symmetrically-spaced pairs of guardelectrodes, symmetrically-spaced pairs of voltage monitor electrodes,and a pair of return electrodes, the voltage monitor electrodes beingpositioned in dyads between adjacent ones of the center electrode andguard electrodes, each dyad being two symmetrically-spaced pairs forminga set of four monitor electrodes that includes, on each side of thecenter electrode, two monitor electrodes between said adjacentelectrodes; respective current sources that drive the center electrodeand each pair of guard electrodes relative to the pair of returnelectrodes; one or more analog to digital converters that digitizerepresentations of differential voltage signals between said dyads; andat least one processor that processes said digitized representations todetermine a resistivity measurement associated with a tool having areduced number of guard electrodes.
 18. The tool of claim 17, comprisingat least five symmetrically-spaced pairs of guard electrodes.
 19. Thetool of claim 17, wherein the current sources drive all of said primaryand guard currents concurrently.
 20. The tool of claim 19, wherein eachcurrent source provides a current with a spectral signature thatdistinguishes it from the currents from the other current sources.